Vector space/Change of base field/Properties/Fact

Let ${}K$ be a field, ${}V$ be a ${}K$-vector space, and ${}K\subseteq L$ a field extension. Then the following statements hold.

1. The tensor product ${}L\otimes _{K}V$ is an ${}L$-vector space.
2. There exists a canonical ${}K$-linear mapping

   $V\longrightarrow L\otimes _{K}V,v\longmapsto 1\otimes v.$

   For ${}K=L$, this is an isomorphism.

3. For a ${}K$-linear mapping ${}\varphi \colon V\rightarrow W$, the induced mapping

   $\operatorname {Id} _{L}\otimes \varphi \colon L\otimes _{K}V\longrightarrow L\otimes _{K}W$

   is ${}L$-linear.

4. For ${}V=K^{n}$, we have

   ${}L\otimes _{K}K^{n}\cong L^{n}\,.$

5. For a finite-dimensional ${}K$-vector space ${}V$, we have

   ${}\dim _{K}{\left(V\right)}=\dim _{K}{\left(L\otimes _{K}V\right)}\,.$

6. For another field extension ${}L\subseteq M$, we have

   ${}M\otimes _{K}V\cong M\otimes _{L}{\left(L\otimes _{K}V\right)}\,$

   (an isomorphism of $M$-vector spaces).

Proof, Write another proof